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Solitons in low-dimensional sigma models

Abstract

The aim of this thesis is to study topological soliton solutions in classical field theories, called sigma models, on a three-dimensional space. In chapter 1 we review the general field-theoretical framework of classical soliton solutions and exemplify it on the main features of the 0(3) σ-model and the Abehan Higgs model in (2+1) dimensions. In chapter 2 a U(l)-gauged 0(3) σ-model is discussed, where the behaviour of the gauge field is determined by a Chern-Simons term in the action. We find numerical solutions for radially symmetric fields and discuss those of degree one and two. They carry a non-vanishing angular momentum and can be interpreted as classical anyons. A similar model is studied in chapter 3. Here the potential is of Higgs-type and chosen to produce a Bogomol'nyi model where the energy is bounded from below by a linear combination of the topological degree of the matter fields and the local U(l)-charge. Depending on internal parameters, the solutions are solitons or vortices. We study them numerically and prove for a certain range of the matter field's vacuum value that there cannot be a 1-soliton.In chapter 4 we discuss a modified 0(3) σ-model in (3+0) dimensions. The topological stability of the solitons is here imphed by the degree of the map S(^3) → S(^2), which provides a lower boundon the potential energy of the configuration. Numerical solutions are obtained for configurations of azimuthal symmetry and the spectrum of slowly rotating solitons is approximated. Chapter 5 deals with a theory where the fields are maps IR(^2+1) → CP(^2). The Lagrangian includes a potential and a fourth-order term in the field-gradient. We find a family of static analytic solutions of degree one and study the 2-soIiton configuration numerically by using a gradient-flow equation on the moduli space of solutions. We conclude this thesis with a brief summary and give an outlook to open questions.